ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-09-04

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The three transposition subgroups of S_3 are conjugate complements to A_3

Example

In $S_{3}$ , the subgroup $A_{3} = ⟨ (123) ⟩$ has as complements exactly the three order-two subgroups

$⟨ (12) ⟩, ⟨ (13) ⟩, ⟨ (23) ⟩,$

and they are conjugate.

Facts & Assumptions

Given: The symmetric group $S_{3}$ (The symmetric group $Sym (X)$ : the bijections of a set $X$ under composition) and its alternating subgroup $A_{3}$ (The alternating group $A_{n} = \ker (sgn)$ of even permutations).

[L1]

Schur-Zassenhaus gives conjugacy of complements when the quotient is solvable (Schur-Zassenhaus conjugacy when the kernel or quotient is solvable).

Verification

technique · direct

1.1

The subgroup $A_{3}$ has order $3$ and index $2$ , so each subgroup generated by a transposition intersects it trivially and together they generate $S_{3}$ . Thus the three transposition subgroups are complements to $A_{3}$ .

givenalgebra

2.1

They are conjugate by direct calculation: $(123) (12) (123)^{- 1} = (23)$ and $(132) (12) (132)^{- 1} = (13)$ . Since $S_{3} / A_{3} ≅ C_{2}$ is solvable, this also matches [L1].

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

David A. Craven, Finite Group Theory (standard reference, not scraped)